Related papers: Divisors on graphs, orientations, syzygies, and sy…
We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe…
We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gr\"obner theory. We give an explicit description of a minimal Gr\"obner bases for each higher syzygy module. In each…
Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$…
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of…
In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is…
We define an independence system associated with simple graphs. We prove that the independence system is a matroid for certain families of graphs, including trees, with bases as minimal resolving sets. Consequently, the greedy algorithm on…
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…
The emergence of Boij-S\"oderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a k-linear minimal resolution…
There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the…
Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).
Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and…
We study families of cellular resolutions by looking at them as a category and applying tools from representation stability. We obtain sufficient conditions on the structure of the family to have a noetherian representation category and…
This paper studies the zero-divisor graphs attached to several finite chain-ring families and computes the homological invariants of their edge ideals by using cochordal constructible systems. We begin with a general layered graph $C(q,L)$,…
A celebrated theorem of Fr\"oberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active…
We formulate a conjecture on the behavior of the minimal free resolutions of sets of general points on arbitrary varieties embedded by complete linear series, in analogy with the well-known Minimal Resolution Conjecture for points in…
Based upon a previous work of Manjunath and Sturmfels for a finite, complete, undirected graph, and a refined algorithm by Er\"ocal, Motsak, Schreyer and Steenpa{\ss} for computing syzygies, we display a free resolution of the lattice ideal…
We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications,…
We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a…
We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with…
In this thesis we investigate certain types of monomial ideals of polynomial rings over fields. We are interested in minimal free resolutions of these ideals (or equivalently the quotients of the polynomial ring by the ideals) considered as…